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3330_Notes_4o4_fill - Math 3330 Section 4.4 Cosets of a...

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Math 3330 Section 4.4 Cosets of a Subgroup Part 1 : Products of Subsets – Cosets Let A and B be nonempty subsets of the group G. Define the product of AB to be the subset { } | ,for some , AB x G x ab a A b B = = Examples: Properties (AB)C = A(BC) AB is not necessarily the same as BA AB = AC does NOT imply B = C. Example:
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For an element g in the group G, gA = gB implies A = B Why? This concept is most useful if at least one of the sets A or B is a subgroup of G. One special version is one subset has a single element in it and the other is a subgroup. Cosets Let H be a subgroup of the group G. For any element a of the group G, { } | ,for some aH x G x ah h H = = Is a left coset of H in G. Similarly, Ha is called a right coset of H in G. Note - aH and Ha are never disjoint subsets of G. Why? Examples: G =
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